Yun Li and Fan Yang

Abstract—In this paper, an adaptive backstepping control scheme is proposed for attitude tracking of non-rigid spacecraft in the presence of input quantization, inertial uncertainty and external disturbance. The control signal for each actuator is quantized by sector-bounded quantizers, including the logarithmic quantizer and the hysteresis quantizer. By describing the impact of quantization in a new affine model and introducing a smooth function and a novel form of the control signal, the influence caused by input quantization and external disturbance is properly compensated for. Moreover, with the aid of the adaptive control technique, our approach can achieve attitude tracking without the explicit knowledge of inertial parameters. Unlike existing attitude control schemes for spacecraft, in this paper, the quantization parameters can be unknown, and the bounds of inertial parameters and disturbance are also not needed. In addition to proving the stability of the closed-loop system, the relationship between the control performance and design parameters is analyzed. Simulation results are presented to illustrate the effectiveness of the proposed scheme.

I. Introduction

IN recent years, the fractionated/plug-and-play spacecraft has received increasing attention in aerospace engineering.Since being proposed in 2006, several experimental satellites,including PnPSat 1, TacSat, and MSV, have been lunched[1]–[4]. The functional components of this kind of spacecraft are connected via wireless networks. Unlike the traditional cable connected architecture, which is heavy, costly and awkward, the wireless architecture provides possibilities to construct faster, lighter weight, flexible and recyclable spacecraft,which makes this fractionated architecture more advantageous and promising.

A key technology for fractionated spacecraft is wireless signal transmission. As analyzed in [5]–[7], in order to transmit signals in the network, quantization is an inevitable and important process. However, for fractionated spacecraft,the bandwidth of the wireless network, which transmits data from the controller to actuators, is limited. This limitation brings large quantization errors into the control signal, and can cause control performance deterioration or even instability. In addition, spacecraft dynamics usually suffer from environmental disturbance (such as gravitation, magnetic forces, solar radiation, etc.). In order to improve performance and robustness of control systems, the influence of quantization error and external disturbance must be taken into consideration.

As one of the most important subsystems for spacecraft, the attitude control system plays a key role in determining whether spacecraft can successfully accomplish on-orbit missions or not, and even the lifespan of spacecraft. Up to now, extensive research has been done on attitude control of spacecraft, and many effective control approaches have been applied, such as optimal control [8], iterative learning control[9], model predictive control [10], robust control [11],passivity based control [12] and sliding mode control [13].Since inertial parameters are difficult to be precisely determined in pre-flight testing, much attention was paid to deal with attitude control with inertial uncertainty. As an effective approach to deal with system uncertainty, adaptive control technique has been applied to the attitude control of spacecraft with unknown inertial matrix ([14]–[16] and references therein). It should be pointed out that, however, in aforementioned research works, inertial parameters are assumed to be constant or their derivative is negligibly small.It is true that this assumption is reasonable when the changes of system parameters is sufficiently slow, and can reduce difficulties in controller design. Nevertheless, modern spacecraft is usually equipped with deployable appendages,such as solar arrays and antennas. The deployment of these appendages could cause non-ignorable variations in inertial parameters. Hence, the constant inertial parameters assumption in existing papers will not be applicable under this circumstance. In [17], the influences of mass displacement caused by appendages deployment and fuel consumption are formulated, respectively, and two adaptive control schemes were developed for the attitude control of spacecraft with time-varying inertial matrix. In [18], an adaptive fault-tolerant attitude control is proposed for deployable spacecraft with prescribed tracking performance.

It should be pointed out, although aforementioned attitude control schemes could achieve satisfactory control performance, none of them consider the input quantization issue. Notable exceptions are [19]–[23]. Spacecraft attitude stabilization with logarithmic-type control torque quantization is researched in [19], where the inertial matrix is assumed to be constant, and the designed adaptive law is not robust to uncertainties and disturbances. Without considering external disturbance, attitude control schemes based on sliding-mode control and passivity are developed for spacecraft with input quantization in [20] and [21], respectively. In [22], by utilizing the disturbance observer, a fixed-time attitude control scheme is proposed for spacecraft with quantized control torque. Note that in [20]–[22], inertial parameters are all assumed to be totally available for controller design. In [23],the quantized attitude control for spacecraft with inertial uncertainty and external disturbance is researched. However,in [23], in order to guarantee the closed-loop stability,additional restriction is imposed on quantization parameters;the inertial matrix is required be constant and the bounds of unknown parameters need to be available, which largely restricts the practicality of the control scheme.

In this paper, a novel adaptive backstepping control scheme is proposed for attitude maneuver of spacecraft in the presence of input quantization, unknown external disturbance, timevarying inertial parameters and redundant actuator configuration. The control signal for each actuator is quantized with a class of quantizers satisfying sector-bound property, including the logarithmic quantizer and the hysteresis quantizer. In order to deal with the influence of quantization error, a new affine model is employed to describe the influence of quantization, and a smooth function, as well as a novel form of designed control signals are introduced.Unlike existing attitude control approaches for spacecraft with quantized input, the quantization parameter can be unknown,and no information about the bounds of inertial uncertainty and disturbances are needed. Besides, the redundant actuator configuration and deployable appendages induced timevarying inertial parameters are also considered.

The remaining parts of this paper are organized as follows.Section II states the problem formulation. Section III gives the design procedures of the adaptive backstepping controller,followed by the stability analysis in Section IV. Section V presents the numerical simulation to verify the effectiveness of the proposed control scheme.

II. Problem Formulation

A. Attitude Dynamics

The nonsingular attitude kinematics and dynamics of the non-rigid spacecraft can be described in terms of unit quaternion as [17]

The attitude control system over wireless communication for spacecraft with input quantization is presented in Fig. 1.The control signalis quantized in the quantizer, and then coded in the coder. After being transmitted via the communication network, the quantized signal is recovered in the decoder, and further is applied to actuators. We assume that the communication network is noiseless so that the quantized signal can be fully recovered in the decoder.

Fig. 1. The attitude control system for spacecraft with quantized control input.

In this paper, we consider that the spacecraft is equipped with deployable appendages, which can result in appreciable variations in inertial matrix, and the time-varying inertial matrix model studied in our upcoming section is modeled as

Assumption 1:The external disturbancesatisfieswithan unknown constant.

Assumption 2:The inertial moment matrixis symmetric and positive definite.

Remark 1:Compared with the assumption of bounded disturbance in [11], [16], [18], [23], the Assumption 1 is more general and encompasses more kinds of disturbance, e.g., the aerodynamic drag which is proportional to the square of angular velocity. Assumption 2 is pretty common in existing works, including those found in [8], [10], [12], [14] and[16]–[23].

In order to formulate our control problem, we defineas the relative attitude error betweenand desired attitude orientationwhich can be computed as

Note that, to guarantee the controllability of the attitude control system, the distribution matrixDis made full row rank by proper displacement of actuators. Attitude and attitude velocity are assumed to be measurable.

B. Quantizers

This paper considers two types of quantizers: the logarithmic quantizer and the hysteresis quantizer. Both kinds of quantizers satisfy a so-called sector-bound property, which will be illustrated in the following.

1) Logarithmic Quantizer:The logarithmic quantizer can be modeled as

2) Hysteresis Quantizer:The hysteresis quantizer can be formulated as

式(11)為t階段船舶貝內(nèi)橫傾力矩約束，其中：HM為船舶貝最大允許橫傾力矩，為船舶貝的列數(shù)；d為集裝箱寬度；Δ為相鄰集裝箱間隙；δ為力矩敏感系數(shù)；g為重力加速度。

Fig. 2. The map of quantizer for .

Remark 2:As can be seen from (9) and (10),determines the size of the dead-zone forThe parametercan be regarded as a measurement of the quantization density. The smalleris, the finer the quantizer is, and the number of quantization levels increases, which can result in a more precise control, but with more communication quantity needed. Compared with the logarithmic quantizer, the hysteresis quantizer has more quantization levels that can reduce the chattering phenomenon, because a dwell time exists when thein Fig. 2(b) transits from one value to another.

Lemma 1:For the logarithmic quantizer (9) and the hysteresis quantizer (10), we have the following relationship:

Lemma 2:The control gainand the additive disturbancelike termhave the following properties:

Proof:With the sector-bound property of the quantizers (9)and (10), it can be derived thatforandwhich further givesforAlong with the relationship in (12), it is obvious thatWith the definition of quantizers in (9) and (10), we can obtain thatwhenwhich justifies (13). ■

The control objective in this paper is to propose an adaptive control scheme for attitude tracking of spacecraft in the presence of input quantization, inertial uncertainty and external disturbance such that all signals of the closed-loop system are bounded and the attitude error can be steered into an adjustable residual set. For this end, the following lemma is needed.

Lemma 3 [24]:For any scalarand constantthe following inequality holds

III. Controller Design

In this section, an adaptive control scheme is deduced within the backstepping framework. The design procedure consists of two steps, and the control signal is obtained at the last step. For brevity, we employ positive scalarsandas design parameters without restating.

Step 1:Define the first error variable as

whose derivative by (8) is

At this step, we consider the quadratic form

Along with (16) and (8), calculating the derivative ofyields

Step 2:Define the second error variable as

Taking (8), (3) and (4) into account, we have

With this definition, (22) can be rewritten as

According to Lemma 1, Assumption 1 and the H?lder’s inequality, it follows that

In this paper, we consider that the quantization parameteris unknown in controller design, which is not taken into consideration in [20]–[23]. Sinceis unavailable for controller design, we define

where the undetermined functionwill be designed later.From (11), (12) and (13), it is obvious thatis symmetric positive definite. Further, based on (27), Lemma 3 and the H?lder’s inequality, iffor allwe have

Consider the quadratic function

From (31), it is obvious thatforwhich justifies (28). Based on the Young’s inequality, we have

Using the property of traceandit can be derived that

choose

Substituting (36) and (37) into (35), by direct calculation,we have

Remark 3:It should be pointed that the influence introduced by quantizers is properly compensated for by the decomposition of quantizers (11), the inequality (14) and the novel form of the control signal (27) and (36). Unlike methods in [20]–[23], the proposed control scheme in this paper does not require the explicit information of inertial matrix as well as their bounds. Furthermore, in [20] and [23], in order to deal with the quantization error, control signals are implicitly considered in the design of the control law, which gives rise to additional restriction on quantization parameters and difficulty in stability analysis. In this paper, the control signal is directly related with state variables, estimated parameters and desired attitude, which is easier in implementation. Besides,quantization parameters can be unknown in our paper, which allows the proposed control scheme to be applicable for a larger class of quantizers as long as the sector-bound property holds. Furthermore, except for dealing with the input quantization phenomenon, our proposed control scheme can also handle the input dead-zone nonlinearity, which can be modeled in a similar affine function as in (11).

Remark 4:Compared with the attitude control scheme for spacecraft in [19]–[23], in which only the quantization for three-axis virtual control torque is considered, we consider the attitude control of over-actuated spacecraft, and the control signal for each actuator is quantized, which is more meaningful but difficult in controller design. Besides, the time-varying inertial uncertainties caused by deployable appendages is studied in this work, which the existing quantized attitude control schemes for spacecraft cannot solve directly.

IV. Stability Analysis

Theorem 1:Consider the closed-loop system consisting of the spacecraft system (1), the quantizers (9) and (10), the adaptive laws (31) and (37) and the control signals (27) and(36). Supposed that Assumptions 1 and 2 hold. Then all signals of the closed-loop system are globally bounded, and the attitude error can be steered into an adjustable residual set.

Proof:Define the Lyapunov candidate function for the closed-loop system as

which means that the attitude error can converge to a small residual set by inc reasing a and decreasing b.■

In the following, we will show the transient performance of attitude error in the sense ofandnorm. Based on (17)and (41), it can be derived that

Integrating both sides of (44) gives that

On the other hand, when the initial attitude of the spacecraft is in that of the desired orientation, that isandit is implied thatIf the initial condition ofandcan be set as zero, then we have

Substituting (46) into (42) givesforAs a result, we have

Remark 5:From above analysis, it can be seen that the global stability can be guaranteed no matter how coarse the quantizer is. Note that, however, due to the existence of termthe ultimate bound of the attitude error is proportional to the dead-zone of the quantizerfor the same control parameter. Hence, by choosing a smallerbetter control performance can be achieved.

Remark 6:As for choosing design parametersandwe have the following guideline:

1) The global stability can be guaranteed as long asi=1,2 ,=1,2,3, and ? are positive.

2) According to (41), increasingandhelps increaseand increasingand decreasinghelps decreaseAlthough increasingcan help increaseit simultaneously increaseBased on this fact, we can first fixthen increaseandIn this way, we can increasewithout increasing

V. Simulation Results

In this section. to verify the effectiveness of the proposed control scheme, numerical simulation on MATLAB/Simulink is performed. In the simulation, the expansion of antennas are considered, which results in a time-varying inertial matrixand can be formulated asHere, we set

The deployment process of two pieces of antennas is shown in Fig. 3. Suppose that the centers of two pieces of antennas locate atandrelative to the mass centerrespectively, and the coordinates ofandcan be given as

which means that it takes 10 s for antennas to be fully expanded.

On the basis of parallel-axis theorem [17], the time-varying portion of inertial matrixcan be expressed as

In our simulation, reaction wheels are employed as actuators, and a common configuration of four reaction wheels for spacecraft is given in Fig. 5 . With this configuration, the distribution matrix is

Fig. 3. Mass displacement of spacecraft due to deployment of two antennas.

Fig. 4. Evolution of the diagonal elements of inertial matrix

Fig. 5. Configuration of reaction wheels.

The external disturbance is set asd=0.0005×[1+sin(t/10)+||w||2, 1.5+cos(t/15)+||w||2, 1.2+sin(t/20)+||w||2]T.The spacecraft is steered to track the desired attitude[qd0,qd]Twithandfrom the initial conditionandThe value of design parameters are given in Table I. In the simulation, in order to make a comparision, two cases are considered. In case 1, the logarithmic quantizer (9) is used to quantize the control singaland in case 2, the hysteresis quantizer (10) is utilized toquantizeParameters of quantizers are set asumin=0.0005 andSimulation results are depicted in Figs. 6 and 7.Figs. 6(a), 6(b) and Figs. 7(a) and 7(b) show that our proposed scheme can achieve satisfactory control performance despite the presence of inertial uncertainty, external disturbance and quantization error with both the logarithmic quantizer and the hysteresis quantizer. With the comparison of Figs. 6(c)–6(f)and Figs. 7(c)–7(f), it can be seen that the hysteresis quantizer can alleviate the chattering phenomenon, which is more serious with the logarithmic quantizer.

TABLE I Simulation Parameters of Different Controllers

For the purpose of illustrating the effectiveness of our proposed scheme, using the hysteresis quantizer, the adaptive control scheme proposed in [23] is applied with the same condition, and pseudo-inverse based control allocation is used(namelywithis the pseudo inverse ofandthe designed control torque). The value of design parameters is given in Table I. In the simulation, Scheme 1 represents our proposed control scheme, and Scheme 2 is the adaptive control approach in [23]. The energy consumptions of the two schemes are plotted in Fig. 8 , which is defined asIt can be seen from Figs. 8–10 that by consuming the same amount of energy, our proposed scheme can achieve better control performance.

Further, in order to testify the robustness of the proposed control scheme, 200 Monte Carlo simulations are carried out with respect to stochastic uncertainties in inertial matrix. In the Monte Carlo simulations, the setting is the same as before,except that the inertial matrixis replaced bywith[1+0.1?rand(t)], whereandare the elements of the matrixa ndrespectively. This setting means that the uncertainty of inertial parameters stochastically varies within 10% of their nominal values. The corresponding steady errors of Euler angle and angular velocity vector,which are defined as one third of the largest magnitude of the norm of attitude error and angular velocity error during the last 20 seconds of the simulation, namely||x(t)||, are presented in Figs. 11 and 12, from which it is clear that satisfactory control performance can be maintained using the same controller with the same design parameters in the existence of stochastic inertial uncertainties.

Fig. 6. Simulation results with the logarithmic quantizer.

Fig. 7. Simulation results with the hysteresis quantizer.

Fig. 8. Norm of quaternion error vector

Fig. 9. Norm of angular velocity error vector (rad/s).

Fig. 10. Energy consumptions for Schemes 1 and 2.

Fig. 11. Monte Carlo results with logarithmic quantizer (a) steady state error of Euler angle vector (deg); (b) steady state error of angular velocity error vector (deg/s).

Fig. 12. Monte Carlo results with hysteresis quantizer (a) steady state error of Euler angle vector (deg); (b) steady state error of angular velocity error vector (deg/s).

VI. Conclusion

This paper proposes an adaptive backstepping control scheme for the attitude reorientation of spacecraft subject to input quantization, time-varying inertial parameters, external disturbance and redundant actuator configuration. By describing the impact of quantization with a new affine model and introducing a smooth function, as well as a novel form of control signal, the influence caused by quantization errors is properly dealt with, and the quantization parameters can be unknown. Furthermore, our proposed scheme does not require the explicit information of inertial parameters, and the bounds of inertial uncertainties and disturbance can also be unknown.We have shown that all closed-loop signals are globally bounded, and attitude error can be steered into an adjustable residual set. Simulation results are presented to verify the effectiveness of the proposed control scheme.